NAME _____________________________________________ DATE ____________________________ PERIOD _____________
4-6 Study Guide and Intervention
The Quadratic Formula and the Discriminant
Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Quadratic Formula
The solutions of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, with a ≠ 0, are given by x =
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
.
Example: Solve 𝒙𝟐 − 𝟓𝒙 = 14 by using the Quadratic Formula.
Rewrite the equation as 𝑥 2 – 5x – 14 = 0.
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
Quadratic Formula
=
−(−5) ± √(−5)2 − 4 (1)(−14)
2(1)
Replace a with 1, b with –5, and c with –14.
=
5 ± √ 81
2
Simplify.
=
5±9
2
x=
= 7 or –2
The solutions are –2 and 7.
Exercises
Solve each equation by using the Quadratic Formula.
1. 𝑥 2 + 2x – 35 = 0
2. 𝑥 2 + 10x + 24 = 0
3. 𝑥 2 – 11x + 24 = 0
4. 4𝑥 2 + 19x – 5 = 0
5. 14𝑥 2 + 9x + 1 = 0
6. 2𝑥 2 – x – 15 = 0
7. 3𝑥 2 + 5x = 2
8. 2𝑦 2 + y – 15 = 0
9. 3𝑥 2 – 16x + 16 = 0
3𝑟
5
12. 𝑥 2 – 10x – 50 = 0
2
25
10. 8𝑥 2 + 6x – 9 = 0
11. 𝑟 2 –
13. 𝑥 2 + 6x – 23 = 0
14. 4𝑥 2 – 12x – 63 = 0
Chapter 4
+
=0
36
15. 𝑥 2 – 6x + 21 = 0
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
4-6 Study Guide and Intervention (continued)
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant
The expression under the radical sign, 𝑏 2 – 4ac, in the Quadratic Formula is called the
discriminant.
Discriminant
Type and Number of Roots
2
𝑏 – 4ac > 0 and a perfect square
2 rational roots
𝑏 2 – 4ac > 0, but not a perfect square
2 irrational roots
2
𝑏 – 4ac = 0
1 rational root
𝑏 2 – 4ac < 0
2 complex roots
Example: Find the value of the discriminant for each equation. Then describe the number and type of roots for
the equation.
b. 3𝒙𝟐 – 2x + 5 = 0
The discriminant is 𝑏 2 – 4ac = (−2)2 – 4(3) (5) or
–56. The discriminant is negative, so the equation has
2 complex roots.
a. 2𝒙𝟐 + 5x + 3 = 0
The discriminant is 𝑏 2 – 4ac = 52 – 4(2) (3) or 1. The
discriminant is a perfect square, so the equation has 2
rational roots.
Exercises
Complete parts a–c for each quadratic equation.
a. Find the value of the discriminant.
b. Describe the number and type of roots.
c. Find the exact solutions by using the Quadratic Formula.
1. 𝑝2 + 12p = –4
2. 9𝑥 2 – 6x + 1 = 0
3. 2𝑥 2 – 7x – 4 = 0
4. 𝑥 2 + 4x – 4 = 0
5. 5𝑥 2 – 36x + 7 = 0
6. 4𝑥 2 – 4x + 11 = 0
7. 𝑥 2 – 7x + 6 = 0
8. 𝑚2 – 8m = –14
9. 25𝑥 2 – 40x = –16
11. 6𝑥 2 + 26x + 8 = 0
12. 4𝑥 2 – 4x – 11 = 0
10. 4𝑥 2 + 20x + 29 = 0
Chapter 4
37
Glencoe Algebra 2